@Dev-iL, why not ask that on the main site? It'll get more attention there, you'll have formatting & other features available, & the answers will help people with similar questions in the future.

@Dev-iL You will need to be more specific about what you are referring to. Generically, systems of N equations in K unknowns define manifolds of dimension max(0,K-N), suggesting K-N is the relevant number, not N. Also, it's unclear how you are conceive of using "measurements" to reach an optimum of some objective function on that solution manifold.

@whuber Thanks for your comment! One can think of this as a parameter estimation problem, where the optimal solution minimized the SSE of the system of equations. To be more specific, I have 3 unknown and M measurements (equations), and I'm looking for ways to justify that M of 3 or 4 is sufficient. As mentioned, the data is practically noiseless, but there is some unknown degree of correlation between the measurements. One of my issues is that I'm not sure how to state this problem formally.

Not even "formally" so much as "clearly enough". I'll know I'm on the right path when I can at least explain my problem in a non-ambiguous manner to the people in this channel :)